Maximum power output of multistage irreversible heat engines under a generalized heat transfer law by using dynamic programming

A multistage irreversible Carnot heat engine system operating between a finite thermal capacity high-temperature fluid reservoir and an infinite thermal capacity low-temperature environment with a generalized heat transfer law [q∝(Δ(Tn))mq∝(Δ(Tn))m] is investigated in this paper. Optimal control theory is applied to derive the continuous Hamilton-Jacobi-Bellman (HJB) equations, which determine the optimal fluid temperature configurations for maximum power output under the conditions of fixed initial time and fixed initial temperature of the driving fluid. Based on the universal optimization results, the analytical solution for the case with Newtonian heat transfer law (m=1,n=1m=1,n=1) is further obtained. Since there are no analytical solutions for other heat transfer laws, the continuous HJB equations are discretized and the dynamic programming (DP) algorithm is performed to obtain the complete numerical solutions of the optimization problem. Then the effects of the internal irreversibility and heat transfer laws on the optimization results are analyzed in detail. The results obtained can provide some theoretical guidelines for the optimal design and operation of practical energy conversion and transfer processes and systems.